Questions tagged [computational-number-theory]

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7 questions
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generating function of sum of divisors function

It is well known that the function $$\sigma_k(n)=\sum_{d|n}d^k$$ has a generating function. For a number field $K$, suppose that $\mathfrak{a}, \mathfrak{b}$ are ideals in some ideal class $C$ and ...
12 views

how to split a separable algebra?

I'm trying to factor ideals in a function field (more precisely, ideals in a maximal order of a function field), and I've come across a step in the published Buchman-Lenstra algorithm which works in ...
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Probability the Fermat test returns “probably prime”

We aim to show that probability of odd $n>1$ passing the Fermat test for all bases a coprime to n is $$\frac{1}{\phi(n)}\prod_{p|n, p \ prime}gcd(p-1,n-1)$$where $\phi$ is the Euler totient ...
13 views

Determine the set of integers that are represented by the binary quadratic form (1,0,-1) [duplicate]

I need help with finding the set of integers represented by the form (1,0,-1). This is essentially f(x,y) = x^2 - y^2 which can be factorised into (x + y)(x - y) and the determinant is d = 4 > 0.
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Testing whether two number fields are isomorphic

Could somebody point me to an implementation of an algorithm that takes two number fields $F$ and $L$, tests whether they are isomorphics, and if they are, returns an isomorphism? There is such a ...
Orbits of the action of $\mathrm{SL}_{n} (\mathcal{O}_{K})$ on $\mathbb{P}^{n-1}(K)$
Let $K$ be a number field, and let $\mathcal{O}_{K}$ be the ring of integers of $K$. Consider the natural action of $\mathrm{SL}_{n} ( \mathcal{O}_{K})$ on $\mathbb{P}^{n-1}(K)$. It is not difficult ...
In a recent press release off the Great Internet Mersenne Prime Search distributed computing project page, it is announced that $$2^{82589933} - 1$$ is the largest known (Mersenne) prime, ...